I’m afraid you won’t find any dinosaur fossils in Vestal, or in fact in most of New York State, except in the museums! The rock exposed here in Broome County is all from the Upper Devonian period, approximately 380 million years old (130 million years before the dinosaurs appeared). At that time Vestal and most of New York was covered by a shallow tropical sea, so the fossils we find here in our rocks are almost all marine creatures (brachiopods, mollusks, crinoids, etc.) and the occasional plant fragment that had floated in from the land mass to the east.
Dinosaurs did undoubtedly wander across New York State later, during the Mesozoic era (250 to about 65 million years before present). New York was dry land then, and must have hosted many different life forms including the dinosaurs. However, any sediments that did get deposited then and changed to rock have long since disappeared in New York (but not, for instance, in the western United States). They were eroded away over the years since then by wind, rain, flowing water, and lastly the scraping of the glaciers during the last ice age.
The one exception to this is in the “Newark Basin” in southeastern New York. There, some red sandstones representing the shores of lakes in the Late Triassic Period (about 230 million years ago) are still preserved. And this is where we find the three-toed footprints of Coelophysis, the one dinosaur we have evidence for in New York. We know quite a bit about Coelophysis because many skeletons of this dinosaur have been found, though mostly in the western states such as New Mexico. Coelophysis was a small dinosaur, up to 9 feet (3 meters) in length from the tip of its tail to tip of its nose. It was slightly built, with a long thin neck and tail, and had short arms with three-fingered hands. It was bipedal, that is, it walked on its hind legs. Coelophysis was a carnivore, and possibly a scavenger. We know this from finding Coelophysis fossils with the bones of its last meal preserved where its stomach would be.
You can see a reconstruction of Coelophysis at the Museum of the Earth in Ithaca, New York (web site: http://www.museumoftheearth.org/). In fact, the Coelophysis is the emblem of the Museum of the Earth.